1)Write an equation for the parabola whose vertex is at (-8,4) and passes through (-6,-2)
answer=y=-3/2(x+8)^2+4
2)Solve x^2>_2x+24
answer=x<_-4 or x>_6
#1, ok
#2, ok
Both answers are correct:
1) The given equation y = -3/2(x+8)^2 + 4 represents the parabola with vertex (-8, 4) and passing through the point (-6, -2).
2) The inequality x^2 ≥ 2x + 24 has solutions x ≤ -4 or x ≥ 6.
To find the equation for a parabola, we need to use the vertex form equation, which is given by: y = a(x - h)^2 + k, where (h,k) represents the vertex of the parabola.
Given that the vertex is (-8, 4), we can substitute these values into the equation: y = a(x + 8)^2 + 4.
To find the value of 'a', we can use the fact that the parabola passes through the point (-6, -2). Substituting these values into the equation, we have: -2 = a(-6 + 8)^2 + 4.
Simplifying this equation, we get: -2 = 4a + 4.
Moving the constant term to the right side, we have: -2 - 4 = 4a.
Simplifying further, we get: -6 = 4a.
Dividing by 4, we find that a = -3/2.
Substituting this value of 'a' back into the equation, we have:
y = -3/2(x + 8)^2 + 4.
Therefore, the equation for the given parabola with vertex (-8, 4) and passing through (-6, -2) is y = -3/2(x + 8)^2 + 4.
Moving on to the second question:
To solve the inequality x^2 ≥ 2x + 24, we need to find the values of 'x' that satisfy this inequality.
First, we can rearrange the inequality to have zero on one side: x^2 - 2x - 24 ≥ 0.
Next, we need to factorize the quadratic equation: (x - 6)(x + 4) ≥ 0.
This means that either both factors are greater than or equal to zero, or both factors are less than or equal to zero.
So, we have two cases to consider:
1) (x - 6) ≥ 0 and (x + 4) ≥ 0.
For the first inequality, x - 6 ≥ 0, we find that x ≥ 6.
For the second inequality, x + 4 ≥ 0, we find that x ≥ -4.
Therefore, for this case, the solution is x ≥ 6.
2) (x - 6) ≤ 0 and (x + 4) ≤ 0.
For the first inequality, x - 6 ≤ 0, we find that x ≤ 6.
For the second inequality, x + 4 ≤ 0, we find that x ≤ -4.
Therefore, for this case, the solution is x ≤ -4.
Combining both cases, we find that the solution to the inequality x^2 ≥ 2x + 24 is x ≤ -4 or x ≥ 6.